Second-order logic : ontological and epistemological problems

Abstract

In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether (a particular form of) second-order logic can count as logic in some (further to be specified) proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not have the "epistemic innocence" which would be needed for, e.g., foundational programmes in (the philosophy of) mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense.